Reflection Grating Couplers at Oblique Incidence

Vol. 4, No. 6/June 1987/J. Opt. Soc. Am. A 1001

Analysis of the reflection grating couplers for waves atoblique incidence

Wee Sang Park and S. R. Seshadri

Department of Electrical and Computer Engineering, University of Wisconsin-Madison,Madison, Wisconsin 53706

Received August 4, 1986; accepted January 15, 1987

A perturbation analysis of a reflection grating for waves at oblique incidence is giveh. Three canonical equationsgoverning the behavior of the reflection grating are systematically deduced, and from them the power-conservationprinciple and reciprocity are established. The characteristics of the output coupler are obtained with particularreference to both the direction and the polarization of the radiated waves. The important features of an inputcoupler are investigated with particular emphasis on the direction and the polarization of the incident wave and thelength of the grating that lead to the maximum of the input-coupling efficiency. The guided wave can have itselectric or magnetic field polarized normal to its sagittal plane, and the grating can be incorporated either on theconducting surface or on the film-cover interface. The relative merits of these four different schemes of operationof the reflection grating are discussed with reference to their applications as output and input couplers.

1. INTRODUCTIONA grounded dielectric film waveguide is used in millimeterand optical wave applications.' For this waveguide, a reflec-tion grating can be designed by incorporating an appropriateperiodic corrugation either on the conducting surface or onthe filmn-cover interface. If the grounding conductor is re-placed by a dielectric substrate, the structure becomes atransmission grating widely used in integrated optics.Many theoretical analyses and experimental observationshave been reported on transmission gratings'-O with refer-ence to such applications as output and input couplers forthe special case of normal incidence for which the propaga-tion vector of the guided wave is parallel to the gratingvector. The treatment of the general case of oblique inci-dence, in which these two vectors are not aligned, is requiredfor the understanding and the design of grating lenses andcurved gratings. For these practical situations, only thecoupling between the guided modes has been investigat-ed.l-' 6 The coupling between the guided mode and theradiated wave in a transmission grating for the case ofoblique incidence was experimentally observed by Ulrichand Zengerle,'7 and its theoretical analysis has been report-ed by Wlodarczyk and Seshadri.1 8

A sagittal plane contains the propagation vector and thenormal to the midplane of the film. Two such sagittalplanes can be introduced-one for the guided mode and theother for the radiated wave. For the special case of normalincidence, the two sagittal planes coincide, and the electro-magnetic problem separates into two simpler problems,namely, one for which the electric vector (transverse electricmode) and the other for which the magnetic vector (trans-verse magnetic mode) is perpendicular to the common sagit-tal plane. For the general case of oblique incidence the twosagittal planes do not coincide, and the radiated wave hasboth polarizations, that is, has components of the electricvector and the magnetic vector perpendicular to the sagittalplane of the radiated wave even when the guide mode hasonly a single polarization, that is, has either the electric fieldor the magnetic field perpendicular to the sagittal plane of

the guided mode. Consequently, the general problem ofoblique incidence is difficult. In a previous paper'9 we pre-sented a perturbation procedure for the analysis of the inter-actions between the guided mode and the radiated wave in areflection grating for the simple case of normal incidence.In this paper we extend the procedure to the general case ofoblique incidence. For optimizing the performance of aninput coupler for plane waves at the normal incidence of theguided mode, the direction of the incident free wave and thegrating length have to be adjusted to proper values to obtainthe maximum input-coupling efficiency. In addition, forplane waves at oblique incidence of the guided mode thepolarization of the incident wave has to be adjusted toachieve the maximum input-coupling efficiency. The opti-mization of the input coupler for plane waves at the obliqueincidence is carefully carried out in this paper.

The paper is arranged as follows. In Section 2 the state-ment of the problem is given. In Section 3 we present aperturbation treatment of one operating scheme of the re-flection grating and deduce the three canonical equationsgoverning the characteristics of the guided mode and theradiated wave. In Section 4 the canonical equations areshown to lead to the power-conservation principle and reci-procity. The general analyses of the output and the inputcouplers are given in Sections 5 and 6, respectively. InSection 7 a design procedure of the reflection grating isexplained and a set of numerical design data is provided.There are four different schemes of operation of the reflec-tion grating, namely, the guided mode can be either a trans-verse electric mode or a transverse magnetic mode and thecorrugation can be either on the conducting surface or on thefilm-cover interface. In Sections 8 and 9, using the numeri-cal data of Section 7, we investigate the characteristics of theoutput and input couplers and examine the relative merits ofthe four different schemes of operation of the reflectiongrating. Conclusions are given in Section 10. In AppendixA we provide the expressions of the coupling coefficients forthe three remaining operating schemes not included in Sec-tion 3.

0740-3232/87/061001-20$02.00 © 1987 Optical Society of America

W. S. Park and S. R. Seshadri

1002 J. Opt. Soc. Am. A/Vol. 4, No. 6/June 1987

2. STATEMENT OF THE PROBLEM

A reflection grating consisting of a dielectric film (efo, i'O), aperfectly conducting substrate, and a dielectric cover (eceo,,Ao) is shown in Fig. 1. A weak periodic corrugation given byx = O(z) is incorporated on the interface of the film and theconducting substrate. By a Fourier analysis the periodicfunction O(z) is expressed as

O(z) = d[ql cos(Kz + 01) + ½0 cos(2Kz + 02) + * * (1)

where d is the average thickness of the film; al (nD) is theamplitude of, and 0 (02) is the phase of, the fundamental(second) harmonic; and zK is the grating vector. It is as-sumed that aj is small compared with unity. We shall con-sider the general case in which the wave vector ft of theguided wave in the reflection grating in the absence of thecorrugation is inclined at the inclination angle Og with re-spect to the grating vector, as shown schematically in Fig. 2.All the fields are assumed to have the harmonic time depen-dence exp(-iwt). We use a normalization that is essentiallyequivalent to taking the speed of light and the intrinsicimpedance of vacuum to be unity. Then the phasor electricfield E and the phasor magnetic field H in each of thedielectric regions satisfy Maxwell's equations and the Helm-holtz equation:

v X E =iwH, (2)

V X H =-ierE, (3)

(V2 + W

2 er)[E, H] = 0, (4)

where Er is the relative permittivity of the dielectric region.The required boundary conditions are that Etan and Htan becontinuous at x = d and Etan vanish at x = O(z).

x

x = d

gatithe o °'o Xt=erae

Fig. 1. Geometry of the reflection grating with the periodic corru-gation on the film-conductor interface.

3. PERTURBATION EXPANSIONS

Since the conducting surface x = O(z) is almost plane, we usea perturbation of the solution corresponding to a planar (x =0) surface. For this purpose, we introduce a formal expan-sion parameter 6 to indicate the smallness of the quantitiesattached to it. We order a, as a and X2 as 62 quantities for thefollowing reasons: al is smaller than unity by 1 order; eventhough X72 vanishes for sinusoidal corrugations, since such aperfect profile is not feasible technically, nonzero X2 is intro-duced as a 2 quantity. The perturbation is carried up toorder 62 by introducing different space scales20 in the z direc-tion and the expansions of E and H as follows:

Z = Z, Z2 = 62Z,

E = Eo + 6E, + 2E2,

H = Ho + 6H, +362 H2-

(5)

(6)

(7)

The fields for each order on the right-hand sides of Eqs. (6)and (7) have x, y, z, and Z 2 as their variables. It is seen fromEq. (4) that each Cartesian component of E and E satisfiesthe Helmholtz equation separately. For convenience, weseparate the total field into two independent modes, namely,a transverse electric (TE) mode for which Ex = 0 and atransverse magnetic (TM) mode for which H, = 0. Theremaining components of E and H can be expressed in termsof E. and H,. If we substitute Eqs. (5)-(7) into Eqs. (2)-(4),and collect terms of the same order of 6, we obtain thefollowing differential equations:

( 2 82 82 2+ + +W E) [E., H.] = 0,

(X2 y2 aZ0

2 E

8x2 r,/Y 8 XYxOaX

a2 + 2 ) = ExEzs E2 xs - i ad Hxs,

/82 2 8 aya2+ W6r) Hy, -irE a Exs + a2HaX2 aZO a~~~xay

(8)

(9)

(10)

(11)

z K

Y +A Fig. 2. Topviewofthe reflection grating. Thowavevectorflofthoguided wave is inclined at an angle 0 with respect to the gratingvector iK.

(2 + W2Er Hz, = iUoEr (9 Exs + a2 Hx,a x2 / 8) axaz0

. = 0, 1;

(12)

0(62):

/82 82 82

ax2 8Y2 8zo2* Er) E 2 , Hx2l = -2 aa [Exo, Hxo],

(13)

a 2 2 y2 = Ex2 + i(o Hx2 + Hxo

kaX2

(A) 'ErjE a9x y ago '(

(14)


c, E, , An_


( d2 + W2E E = a E 2 + d Exo -iw - H.21~&2 2 aXaZ0 8XaZ2 ay 2

( 2 + W2Er)H 2 = -iWer EX2- iWEr z Exo

82+ d Hx 22 (16)

/82 + &) 2E' H 2 = 8WEra EX2 + 82 Hx2 + 2 Hx0.ax27 a~)y 8xaz 0 2 xaZ2

(17)

The boundary conditions at x = d require that for each order3s (s = 0, 1, 2) Ey, Ezs, Hy,, and Hz, be continuous on theplane x = d. The tangential electric field on the surface x =O(z) can be decomposed into two components, namely, Eyand E, + (dx/dz)Ex. Since the surface x = O(z) is nearly theplane x = 0, by expandingE, Ez, and Ex in Taylor series withthe help of Eqs. (1) and (6), one finds the following equiva-lent boundary conditions for each order bs at x = 0:

0(5a) ProblemThe zero-order problem as governed by Eqs. (8)-(12) and(18) and the boundary conditions at x = d corresponds to the

(15) uniform waveguide. For the sake of brevity, we shallpresent a detailed analysis for the case in which the zero-order field is a TM mode, which can be expressed as

Exo = Ngeag(Z 2 )exp(iO,)exp[-j(x - d)]

X exp[ioj(y sin Og + zo cos Og)], d <x < -, (21)

ec cos(kjx)

X= Ngeag(z2)exP(iO) Ef (k d)

X exp[if3j(y sin Ag+ z0 cos 0), 0 < x < d, (22)

where

aej = (2 - 2CY/2,

kj = (W2Ef1 - 12)1/2

(23a)

(23b)

The remaining zero-order field components can be obtainedfrom Eqs. (9)-(12) with Hx0 set equal to zero as follows:

E,0 = iej sin OgExo, d < x < , (24)43j

0(30):

EY0 = 0,

E 0 = 0,

0(3'):

El+ d71 cos(Kzo + 0,) dx EY= 0,

E2, + dn, cos(Kzo + 0,) Ezo

- dilK sin(Kzo + 0,)Exo = 0,

(18a)

(18b)

(19a)

(19b)

0(62):

Ey2 + d-qi cos(Kzo + 0,) dx EY, + d-q2 cos(2Kzo + 02) a EY0

82122Cos 2(KZ0+ 0)- E' =O,+2 dO7 x2

k.E, = i tan(kjx)sin OgExo,

i13j

Ezo = cot OgEYo,

We1

H, = ,B cos OgEo,

H 0= -tan Ogyo,

O<x <d, (25)

(26)0< x < oD,

d <x < -, (27)

(28)O< x <d,

0<x < x. (29)

From Eqs. (18), (25), and (26), the boundary conditions at x= 0 are seen to be automatically fulfilled. The boundaryconditions at x = .d lead to the zero-order dispersion relation

efoaCj = ecki tan(kid).

(20a)

Ez2 + d7, cos(Kzo + 0,) - E, + d-02 cos(2Kzo + 02) Eo8x Ox

+ d2n 2 cos2(Kz0 + 01) - E - 2d72K2 l ax22

X sin(2Kzo + 02)EXo - dqK sin(Kzo + 01)E2,

- 2 d277,

2 K sin(2Kzo + 201) Exo - d 2n 2 K2

2 Ox 0 2

(30)

This relation together with Eqs. (23) always has a solutionfor real fj and positive aj and kj. Hence the lowest TMmode does not have a cutoff frequency. It is seen from Eq.(23a) that fj is greater than o. Consequently, the zero-order field is a slow wave and is referred to as the principalguided wave, whose mode order is indicated by the integer j.The slowly varying amplitude ag(z2) is determined by thecoupling of the principal guided wave to the Floquet modesthat are excited by the periodic boundary conditions on thecorrugated surface. The phase factor exp(iO,) is introducedfor convenience, and the normalization constant Nge is cho-sen as

Nge = [413jefajk j2 /(wD)] 1/2, (31)

where

D = cEf~(k 2 + a 'j2 ) + daxj(E, 2kj 2 + ef2aEXj2). (32)

W. S. Park and S, R. Seshadri

X sin 2(KZO + 01)Eo = 0. (20b)


Then the power Pgz carried by the principal guided wave inthe z direction over a unit width in the y direction is given as

p = lagi2 cos

8 =

(33)

0(61) ProblemThe first-order problem is governed by the differential Eqs.(8)-(12) and by the boundary conditions given in Eqs. (19)and those at x = d. An examination of the boundary condi-tions given by Eqs. (19) with the help of the zero-ordersolutions given in Eqs. (21)-(29) indicates that for the first-order fields (j cos g - K) and (j cos g + K) are the wavenumbers in the z direction and that the wave number in theydirection, 1j sin g, remains unchanged. For the reflectiongrating to be an output or an input coupler, at least one ofthe scattered Floquet modes must be a fast wave. Wechoose the physical parameters of the reflection grating suchthat only [(3j cos Og - K) 2 + 13j2 sin2

Og] 1/2 lies in the fast-wave

region and all the other Floquet modes are slow waves, asshown in Fig. 3. One important feature of the case ofoblique incidence is that in order for the boundary condi-tions to be fulfilled the first- and higher-order fields musthave both the TE- and the TM-mode components regardlessof the mode type of the principal guided wave. Thereforefor the case of oblique incidence we necessarily encounter amode coupling, for which the modes of the principal guidedwave and the radiated wave are of the same type, and a modeconversion, for which the modes of the two waves are ofdifferent types. Hence the solutions of the TM-mode first-order field EX1 and the TE-mode first-order field H.1 aresought in the form

Ex, = Nrejaie exp[-ik 1 ,(x - d)] 4- bieexp[ikjc(x - d)]I

X expi[pj sin 0y + (j cos Og -K)zo]l

+ Fice exp[-ac(x - d)expi[13j sin Ogy

+ (j cos Og + K)z]1, d < x <c, (3

r COs(k-lfx) sin(k-lfx)1LF~fe cos(k-lfd) + fe sin(k-lfd)_

X expli[13, sin y + (j cos Og -K)zo]l

+ F~i cosh(al'x) + G sinh(alfx)cosh (a1 fd) Gfe sinh(a 1 fd)j

X expi[gj sin Oy + (j cos g +

H. 1 = NrIIaih exp[-ikh 1 (x - d)] F bih

- vAYL"J-1ck4 % uIAIJI4UJj bll UY F T jj Lub ug -A40J1

+ Fljh exp[-al(x - d)expi[13j sin Ogy

n = -2

W coW./-j , R~~~~wX

-i

Fig. 3. Brillouin diagram of the Floquet modes scattered by theperiodic corrugation. The hatched and the unhatched regions cor-respond to the fast and the slow waves, respectively. The letter nindicates the order of the Floquet mode whose wave number on theyz plane is given by = [(jcos 0

g + nK) 2+ fj2 sin

2Og]1/

2. n = 0 and

n = -1 denote the principal guided wave and the fast wave, respec-tively, and the Floquet modes n = 1 and n = -2 are slow waves.

F. cos(k-,fx) sin(klfx)1H1=L1h COS(k 1) + - lfh sin(k-lfd)J

X expi[flj sin Ogy + (j COS Og - K)z 0 jI

r cosh(alfx) sinh(afx)+ I~fh + Gfi

L cosh(afd) lfh sinh(alfd)

X expli[#j sin Ogy + (j COS Og + K)zo]},

0 < x < d, (37)

where

kir = ( 2 er - j2 - K 2 + 2K13j cos ag)1/2,

alr = ( 2 + K 2 + 2K,3j COS g - W2E )1/2, r = f, c.

(38a)

(38b)

The upper and the lower signs in Eqs. (34) and (36) corre-spond to the cases for which the principal guided wave is ofthe TM and the TE mode, respectively. The subscripts eand h are used to denote the TM and the TE modes, respec-tively. The solutions for Ex1 in Eq. (34) and H,1 in Eq. (36)are in the form of two Floquet modes, which are fast waves,and a noneigen guided mode, which is a slow wave carryingenergy along the waveguide. The normalized amplitudes aie(ail,) and bie (bih) of the two fast waves for the TM (TE) modecorrespond to the incident and the radiated waves, respec-tively, and ae (aih) is assumed to be known. The normaliza-tion factors Nre and Nrh are chosen as

Nre = [2(j2 + K2 - 2K,3j COS Og)/(cEc*kc)] 112,

Nrh = [2(gj2 + K2 - 2K13j COS Og)/(Wkck_)]1/2.

(39)

(40)

K)z0]j, Then the -x and +x components of the Poynting vectorsS, and S associated with the incident and the radiated

0 <x <d, (35) waves of the TM (TE) mode become lai12(laihl2) and biel2(Ibih12), respectively. Other first-order field components,namely, Ejl, Ez, H,,, and H,,, can be determined from Eqs.(9)-(12) with the substitution of Eqs. (34)-(37). Applica-

a __ a _ T 1 tion of the boundary conditions given by Eqs. (19) yields

G-lfe =dn1ec,[#(kj2- K2 ) + K cos g(#j2 -k 2 )]

2113 1 cos(kJd)k-lf csc(k-lfd)

Ngeag,

(j COS g + K)zo]j, d<x < ,


1

,., ___rn. f- - 14N1IIf.Ir/? - D - __L I

(36) (41)


idnlcK(j 2 + k12 )sin NgNa

-2WoEfo3 cos(kjd)sec(k-lfd) ge g

Gif _ dnE (kj 2 -K2) - K cos Og(j 2 - kj2)]

2,f2efj cos(kjd)alf csch(alfd)

X exp(i201 )Ngeag,

-id 1 lEcK(312 + kj 2 )sin 0 exp(i20 l)Ngag.

fh = 2 .Ef1 cos(kid)sech(alfd)

(42) aie' aih

(43)

(44)

X

biI b ih

Using the first-order boundary conditions at x = d, we deter-mine the remaining amplitudes of the first-order fields inEqs. (34)-(37) as

bie = Crgeag + Crreaie, (45)

bih = Crghag + Crrhaih, (46)

where

Crg = d,7)lEJj(k j2 - K2) + K coS Og(1j2 - kj 2 )]Nge

Crge 213j cos(kd)cos(k-lfd)[ek-lf tan(k-lfd) + iEfklc]Nre

(47)

Crre --ECk.lf tan(klfd) + ifklc (48)k.lf tan(k.lfc) + iEfklC

C = dnlEK(3j 2 + kj 2 )k-lf csc(k-lfd)sin OgNge, (49)rgh

2Efj cos(kjd)[k-1 , + iklf cot(k-lfd)]Nrh

ik-lf cot(k-lfd) + k-le

and

F. _ iEck-lf cot(k-lfd) + Efk. Gl-lfe iEk-lf tan(k-lfd) - Efk-l -

+ Ejk.1~ 2 Eck.l4 Areaie-ifk-,-iEk-lftan(k-lfd)

G-lfh = ik1 lf tan(klfd) - k- 1 F-lfh

ih=lf cot(k-lfd) + k-1 ,

+ 2 k-lcNrhaihik-lf cot(k-1 fd) + k-1 ,

Flee = Efalf[tanh(alfd) - coth(a1 fd)]efalc + ecalf tanh(aolfd)

Fie= efalc + Ealf coth(alfd) GfEfalc + ECElf tanh(alfd)

alf[coth(alfd) + tanh(alfd)Fr - =- I

a1 f coth(alfd) + a,,

yFig. 4. Diagram of the wave vectors in the direct operation of thereflection grating: of the principal guided wave, Or of the radiatedwave, and 13i of the incident wave. The line AB is the projection of hiand Or on the yz plane. The angles Or, Oir, Oi, and 0i are measured asindicated by the arrows.

All the coefficients necessary to describe the first-order fieldcomponents have been determined, and they reduce correct-ly to the corresponding expressions for the special case ofnormal incidence when 0

g = 00.21 Equation (45) [(46)] is arelation between the principal guided wave and the incidentand the radiated waves of the TM (TE) mode. For thepresent analysis, in which the principal guided wave is of theTM mode, Crge is a mode-coupling coefficient and Crgh is amode-conversion coefficient. The mode-conversion coeffi-cient always vanishes when 0

g = 0°.Even though the first-order radiated or incident wave

consists of the TM and the TE modes with reference to the xdirection, it can be verified that sum of the two modes is oftransverse electromagnetic mode with reference to the prop-agation direction. In view of Eq. (34) or Eq. (36) the wavevectors 1 3

r and 3,i of the radiated and the incident waves aregiven by [k 1 l, + yOj sin Og + z(Oj cos Og - K)] and [-Tkhl, +513j sin g + Z(O3j cos g - K)], respectively, and the planeformed by either of these two vectors and a vector normal tothe yz plane is known as a sagittal plane (Fig. 4). If the totalelectric field Er of the radiated wave is decomposed into twocomponents such that Er = Ea, + Evans where #r ap = r -an = a an = 0, and a, and an are unit vectors parallel andnormal to the sagittal plane, then the following relations canbe deduced:

E = bie(2W/klc)1/ 2 , (57)

(52) En = bih(2/kc)/ 2 . (58)

The above two relations and the fact that Ea, and Enan areorthogonal to each other indicate that the two radiated fields

Glfe, (53) represented by bie and bih are also orthogonal to each other.Moreover, the x components of Poynting vectors associatedwith the electric fields Ea, and Enan can be shown to be

(54) 1bie12 and bih 12, respectively. The same results can be ob-tained for the incident wave. Hence we shall use the fouramplitudes a, ah, bie, and bh in expressing the first-order

Il fast waves.'F. fh,

h= - f tanh(alfd) + ale Flfh.Glfh =-alf coth(alfd) + ae1

(55)

0(32) Problem IThe solvability condition of the inhomogeneous second-or-

(56) der problem, as we shall establish now, is the amplitudetransport equation governing the nature of the interaction


- c


between the principal guided wave and the first-ordedent and radiated waves. To determine the solvabilitdition we seek particular solutions for E 2 and HX2 form

EX2 = k2c(x)exp[ij(y sin Og + z cos Og)], d <x

EX2 = k.f(x)exp[i3j(y sin Og + z cos Og)], O <x <

HX2 = 4xc(x)exp[ij(y sin Og + z cos Og)], d <x

r inci- The particular solutions of the remaining second-order fieldy con- components are determined by the substitution of Eqs. (65)-in the (68), (20), and (21) into Eqs. (14)-(17). From the boundary

conditions, given by Eqs. (20), the amplitudes C2e and B2hcan be determined. However, for the derivation of the am-

<, ' plitude transport equation for the principal guided wave of(59a) the TM-mode type we need only C2e as given by

s d, dr/l410 sin(kjd) (59b) C2 e = d 2ik sin, kd exp(i0j)

2i

(60a)

HX2 = O.,f(x)exp[ij(y sin Og + zo cos Og)], O <x <d.

(60b)

Substituting Eqs. (59) and (60) into Eq. (13) with H2 0 = 0 forthe present analysis, we obtain the following differentialequations:

(I - 2 = N d exp(i0j)~ dX2_ acjOxc =-2i3 1 COS gN Z

X exp[-acj(x - d)], (61)

x ik 1,f (flj - K cosg) - i cos og(3j2 + K2 - 2j 1K cos0 g)

L cos(k 1lfd)(1j32 + K2 - 2jK cos Og) J

- G-lfh exp(i0j)WKk-lf sin g

sin(k-lfd) (1j2 + K2 - 213jK cos Og)

+ Fife exp(-iO)

Fr-ia1 7f2(3 + K cos Og) + iK cos g(#j2 + K2 + 2jK cos Og) 1

[ cosh(a 1 fd)(137 + K2 + 2jK cos Og) J

d2

(dx2

(x- cj )xc = ^(dx2 +J)x/ 0O

d 2 2 = -0(dx2 i f

+ Glfh exp(-i0 1)wKalf sin Og

sinh(alfd)0(13:2 + K2 + 213jK cos Og)J(69)

Using the boundary conditions at x = d for the second-order(63) fields, we can deduce that

(64)

The general solutions of Eqs. (61)-(64) satisfying the radia-tion condition at x = are as follows:

Oxc(x) = i cos OgNg e exp(i0j) x

X exp[-acj(x - d)]+ A2e exp[-cj(x - d)],

dagexsnkxox (x) = -ij cos OgNg dg exp(i0) ex sin(kjx)

dZ2 efkj cos(kjd)

B 2 e cos(kx) C sin(kjx)cos(kid) 2e sin(kid)

Axc(x) = A2h exp[-a(x -d)],

cos(k x) sin(kx))/ X(x)B B c J~kd + inkC

EcA2e- fB 2e = -ieco13 cos Ogd I + I tan(kid)

daX Nge de exp(i0j) + EfC2 e,

acj(EcA2e - efB2e) = -ic13j cos g (d - 1 Ef ac)

(65)

(70)

X Nge dae exp (i01) - ckj cot(kjd)C2 e.dZ2

(71)

(66) Eliminating C2e and (cA 2e - fB2e) from Eqs. (69)-(71) leadsto the solvability condition or the amplitude transport equa-tion

(67)

(68)where

d=+ (72)

Cg = d I (e 2 k 2 + Ef2a2)(pl + P2 + P3 + P4),

p [3(kj2 -K2) + K cos Og(g3j2 - kj2)]2[iEck.lf cot(klfd) + fk_C]

k..f cot(k.lfd)(3j 2 + K2 - 2OjK cos 0g)[ieckjf tan(klfd) - fk_.jj

-K2(/ 2 + kj2)kf[ikjf tan(kfd) - kjjsin 2 gP = -- - -c

(,Bl/ +,K2 2jK cos Og) [ik-If + k1, tan(k-lfd)]

2 ~~~dag ccskx+ k 2 Oxf = -2i3 1 cos 0gNg - exp(i0j) c (kjd)ii dz 2 Ef COS(62d)

(62)

(73)

(74)

(75)

W. S. Park and S R Seshadri


-[13(k 2 - K2) - K cos Og(j 2 - kj 2)]2[ECalf coth(ald) + Efal']P3 = (0

Lr.lf coth(aifd)(13 2 + K2 + 21jK cos Og) [calf tanh(aifd) + efaiC]

- K2(j 2 + k12)alf[alf tanh(aifd) + al]sin 2 og

(1 2 + K2 + 21jK cos Og) [alf + a1 c tanh(afd)]

dnEkiki_[1J(kj 2 - K2) + K cos og(3j2 - kj2)](Ef2 aCj2 + Ec2kj2)sin(kjd)NreC91 = 02+K

gre - E .[Eck-lf tan(klfd) + iffk.1c.]J 2 + K2 - 213jK cos Og)cos(k_1fd)DNge

dniwKkf(E 2 k 2 + Ef2a Cj2)sin(kjd)k_1ck_1f sin OgNrh

g [ik.-f cot(klfd) + k- 1 ],(12 + K2- 21jK cos Og)sin(k_1fd)DNge

Equations (45), (46), and (72) constitute the three general-ly valid canonical equations for a six-port distributed cou-pler with a guided wave, two modes of the incident wave, andtwo modes of the radiated wave. The characteristics of thedistributed coupler depend entirely on the seven coeffi-cients, namely, Cgg, Cgre, Cgrh, Crge, Crre, Crgh, and Crrh. As

will be shown below, Cgg, known as the extinction coefficient,is the most important coefficient, and its real part gives theleakage coefficient. It is verified from Eqs. (73)-(77) thatthe leakage coefficient is a function of k-1 ,, but not of alc,and that the imaginary part of the extinction coefficient is afunction of both of them. This indicates that only the fastwave of the first order affects the leakage coefficient but thatboth the fast and the slow waves of the first order affect thephase of the principal guided wave.

4. POWER CONSERVATION ANDRECIPROCITY

The following seven properties can be derived from the defi-nitions of the seven coefficients:

side of Eq. (81) as the net power entering the region throughthe x = plane, which, by definition, is given as

(Iaie 2 + laihl)dz 2 - (bieI2 + Ibil2

)dZ 2. (82)

On substituting Eqs. (45) and (46) into Eq. (82) and manipu-lating with the help of Eqs. (80), the right-hand side of Eq.(81) can be shown to be equal to that given in Eq. (82), thusfulfilling the requirement of power conservation.

The three canonical equations for the reflection gratingshould be valid for the reciprocal operation that is represent-ed by taking the complex conjugate of Eqs. (45), (46), and(72). The amplitudes ag*, aie*, aih*, bie*, and bih* corre-spond to the waves propagating in the corresponding reversedirections and are renamed Ag, Bie, Bih, Aie, and Aih, respec-tively (Figs. 4 and 5). Then we obtain the following equa-tions for the reciprocal operation:

Aie = Crge*Ag + Crre*Bie,

Aih = Crgh *Ag + Crrh*Bih,

cos g dAg = Cgg*Ag + Cgre*Bie + Cgrh*Bih.ICrrel = 1,

ICrrhl = 1,

Crge Cgre,

Crgh = Cgrh,

Cgre*Crre = Cgre,

Cgrh*Crrh = Cgrh,

- 2 Re(Cgg) = Cgrel + Cgrh 2.

(80a)

(80b) With the help of Eqs. (80a), (80b), (80e), and (80f), Eqs. (83)and (84) can be rewritten as

(80c)

(80d)

(80e)

(80f)

(80g)

We shall now prove that Eqs. (80) lead to the requirementsof power conservation and reciprocity. Let us consider aregion bounded by the conducting surface, x = c, and theplanes at Z 2 = 0 and Z2 = dz 2. We multiply Eq. (72) by ag*and take the complex conjugate of the result. Integratingthe sum of the two results with respect to Z 2 over 0 to dz2 , wefind that

(lagz 21dz 2 - lag1z2 =o)cos 0g = 2 Re(Cgg)IagI 2

dz 2

+ 2 Re(Cgreaieag* + Cgrhaihag*)dz 2 . (81)

Since Pgz, defined in Eq. (33), is in the z direction, the power-conservation principle enables us to interpret the left-hand

°i *= Big

a h*= Bh

y

Fig. 5. Reciprocal operation of the reflection grating. The recip-rocal configuration is obtained by taking the complex conjugate ofthe waves in the direct operation.

(76)

(77)

(78)

(79)

(83)

(84)

(85)

Bie = CrgeAg + CrreAie,

Bih = CrghAg + CrrhAih'

(86)

(87)

bi *= A1*

bih*=Aih



Substituting Eqs. (86) and (87) into Eq. (85) and using Eqs.(8 0c)-(80g), we find that

-cos g dz = CggAg + CgreAie + CgrhAih.

JLJ bJI52dZ2

= Ia'(0) 2 cos (88)

The new canonical equations, namely, Eqs. (86)-(88), corre-spond to the reciprocal operation of the reflection grating,and the resulting seven coefficients are identical to those forthe direct operation proving the reciprocity. Hence theseven properties given in Eqs. (80) can be considered toreflect the internal consistency of the canonical equations.

5. OUTPUT COUPLER

s = e, h. (95)

Because of the orthogonality between the two radiatedmodes, it follows that

Qt Qoe + Qoh- (96)

Substituting Eqs. (92)-(94) into Eq. (95) and using the prop-erty in Eq. (80g), we obtain that

QO5 = ICrgsI2[exp(2Cggr sec OgL) - 1]/(2Cggr), s = e, h,

(97)

We shall now analyze an output coupler by solving the ca-nonical equations. Since the operation of an output coupleris based on the scattering of a guided wave into a radiatedwave owing to the presence of a nonuniformity along thewaveguide, the incident waves are not involved (Fig. 6),enabling us to simplify the canonical equations as

bi = Crgeag, (89)

bih =Crghag, (90)

dagcos dag = Cggag. (91)

dZ2

We assume that the corrugation exists over a finite length ofthe waveguide, from 2 = 0 to 2 = L, where L is sufficientlylarge in comparison with a wavelength and the end effectstaking place near the 2 = 0 and 2 = L planes are negligible.These assumptions are also used in the subsequent analysisof the input coupler. The solutions of Eqs. (89)-(91) for ag,bie, and b, subject to the terminal condition that ag(z2 = 0)- ago, are obtained as

ag = ago exp(Cgg sec gZ2), (92)

bi = Crgeago exp(Cgg sec gZ2), (93)

bih = Crghago exp(Cgg sec gZ2). (94)

The total output-coupling efficiency Q is defined as theratio of the total radiated power from the grating to theguided wave power at Z2 = 0. However, since both the TMand the TE modes exist in the radiated wave for the presentoblique-incidence case, we also define the output-couplingefficiency for each mode separately as

/b

Z2 =° Z2 =L

Fig. 6. Operation of the output coupler for 0= 00. The hatched

region denotes the exponential amplitude profile of the radiatedwave.

Qo = 1 - exp( 2 Cggr sec OgL), (98)

where Cggr = Re(Cgg). The expression in Eq. (98) for thespecial case of 0g = 0° is identical to that obtained previous-ly.19 Since Cggr is negative definite, Q and Q approach-ICrgs2/(2Cggr) and unity, respectively, as L gets large. Wecan think of Qoe(Qoh) as a mode-coupling (conversion) effi-ciency for the principal guided wave of the TM-mode type.

The total radiated wave from the output coupler is thesum of the TM- and the TE-mode radiated waves that areobtained by substituting Eqs. (93) and (94) into Eqs. (34)and (36), respectively. The polarization of the total radiat-ed wave is characterized by the polarization factor P, definedas

P = Cgrh/Cgre. (99)

In general, P is a complex number and is a function ofvarious physical parameters of the reflection grating, theinclination angle g, and the operating frequency w. Fromthe explicit expressions of Cgre and Cgrh given by Eqs. (78)and (79), we notice that for a fixed o, there exist severalinteresting special values of P as g varies. P = 0, whichoccurs only when 0g = 0, that is, the normal-incidence case,where the radiated wave is linearly polarized and of the TM-mode type. P = c, where the radiated wave is purely of theTE-mode type, indicating a complete mode conversion,which occurs for a value of g between zero and some criticalangle, smaller than 900, above which the radiated wave is notcoupled. For other values of g, P is a complex number,where the radiated wave is an elliptically polarized wave.However, it is verified from the algebraic expression in Eq.(99) that the real part of P is always nonzero, which meansthat the radiated wave cannot be a circularly polarized wave.

One inherent feature associated with the oblique-inci-dence case is that the wave vector flr of the radiated wave isnot on the plane formed by the wave vector of the principalguided wave and a vector normal to the midplane of the film,as illustrated in Fig. 4. In order to give a full description ofthe radiated wave vector fr, two auxiliary angles 0r and kr aredefined such that r is the angle between the yz plane and flr,and qkr is the angle between the z axis and the projection of l3ron the yz plane. These two angles are readily determinedfrom the solutions given in Eqs. (34) and (36) in the first-order problem as

0r = arctan[k 1 ,/(j 2 + K2 - 2Kj cos g)'12], (100)

/r = arctan[flj sin Og/(13j cos Og - K)]. (101)



If the result of the second-order problem, that is, the second-order modification of the wave number in the z directionderived in Eqs. (92)-(94), is included in Eqs. (34) and (36),the two angles 0 r and 'kr undergo some second-order modifi-cations, and the corresponding modified angles r' and fir'are deduced as

Or' = arctan-l[032sin2 g + (fj COS Og-K + Cggi sec og)2]1/2]

(102)

¢r' = arctan[3 sin Og/(1j cos Og - K + Cggi sec fIg)], (103)

where Cggi is the imaginary part of Cgg. If we extend theperturbation procedure to the third order,22 we could findthe second-order correction in the x component of the wavenumber 62k 12 in the cover region. However, without goingthrough the third-order problem we can deduce k2- 2 bywriting the equation of the propagation constants as follows:

(ik- 1 , + 62k_ 12)2 + (ij sin Og)2 + [i(j cos Og - K)

+ 2 C99 sec fg]2 + W2EC = 0, (104)

where the first, second, and third terms correspond to the x,y, and z components, respectively, of the radiation wavevector. The y component does not change because the ge-ometry of the reflection grating is uniform in the y direction.If we neglect 64 quantities in Eq. (104) and use Eq. (38a), weobtain that

k-2 = -(Oj cos Og - K)Cgg sec Og/k-ic. (105)

The imaginary part of k1 2 affects the radiation anglethrough Or' but not through kr', and the corresponding newangle Or' is expressed as

Or" = arctan

X | k-c- (j COS g -K)Cggi sec O/k l. (106)

l[132 sin2 Og + (1j cos Og - K + Cggi sec og)2]1/2J

Hence, with the results of the perturbation analysis up to thesecond order, Or" and Ckr' are the best approximations de-scribing the propagation direction of the radiated wave fromthe reflection grating.

6. INPUT COUPLER

The reflection grating can be used as an input coupler thatexcites a principal guided wave by applying an appropriateincident wave from above the grating. It should be notedfrom the three canonical equations that the incident wavecan be of either the TM or the TE mode, or a combination ofthe two modes, and that the excitation of the principal guid-ed wave is always accompanied by some reradiation throughthe radiated wave, as is the case for a receiving antenna (Fig.7). The propagation direction of the incident wave is fullydescribed in terms of two auxiliary angles fi and do, as illus-trated in Fig. 4. From Eqs. (34) and (36) it is clear that theexpressions for 0i and 0i are identical to those for 0 r and kr

given in Eqs. (100) and (101), respectively.We shall analyze the input coupler for the incident wave

whose normalized amplitudes aie and aih are slowly varyingfunctions of z. For this purpose, the input-coupling effi-

I I

I ~~~I

Z2 =° Z2 =L

Fig. 7. Operation of the input coupler for 0g = 0°. ag is assumed to

be zero at z 2 = 0. b is the reradiated wave.

ciency Qi is defined as the ratio of the power in the principalguided wave at Z2 = L to the total incident power on thegrating. Then Q is written as

lag(L)I2 cos Og

Jo (laiel2 + laihl')dz 2

(107)

We assume that the amplitude of the principal guided wavevanishes at Z2 = 0. The general solution for ag(z2) is thenobtained from the canonical equation (72) as

ag(z 2 ) = exp(Cgg sec 9gz2 )sec Og f exp(-Cgg sec fgZ2')

X (Cgreaie + Cgrhaih)dz2'- (108)

Substituting Eq. (108) into Eq. (107) gives the general ex-pression of the input-coupling efficiency. However, for il-lustrating the characteristics of the input-coupling efficien-cy we restrict our attention to the incident wave, which is auniform plane wave as represented by

ai, = A, exp[i(l + q)Cggi sec fgZ2], s = e, h, (109)

where A, is a complex amplitude and q is an adjustableparameter to permit the variation of the incident anglesaround Oi' and 0i', which are set equal to Or' and okr' given inEqs. (102) and (103), respectively. Then Oi' and 0i' corre-spond to the case for which q is equal to zero in Eq. (109).Let us consider an incident wave consisting of only onemode. The corresponding input-coupling efficiency Qisl isdeduced from Eqs. (107)-(109) as

lCgrsl2 lexp(Cggr sec OgL) - exp(iqCggi sec OgL)12

basi ICggr - iqCggil2 sec OgL

(110)

Examination of Eq. (110) shows that Qis, is an even functionof q and has a maximum when q = O. Simplifying Eq. (110)for q = 0 and renaming it Qis2 yields

I Cgrs12 [1 - exp(Cggr sec OgL)]2

= C Cggr sec 0IL (111)

The maximum input-coupling efficiency Qis2, max and theoptimum condition resulting in this maximum are analyti-cally determined from Eq. (111) as

Qis2, max =-0.40731Cg 12/Cggr (112a)


1010 J. Opt. Soc. Am. A/Vol. 4, No. 6/June 1987 W. S. Park and S. R. Seshadri

at

Cggr sec OgL = -1.2565. (112b)

It is clear from Eq. (80g) that Qis2, max can have a valuebetween zero and 0.8145 depending on the angle Og.

When the incident wave is a mixture of the TM and theTE modes of the form in Eq. (109), the corresponding input-coupling efficiency Qimi and its optimized version Q2 for q- 0 can be obtained by replacing CgraI2 in Eqs. (110) and(111) by

ICgreAe + Cgr,,AhI2 /(lA I 2 + IAh 2 ), (113)

respectively. It is interesting to note from Eqs. (72) and(108) that the excitation of the principal guided wave can benullified by applying some additional incident wave to thegrating such that the total incident wave satisfies the condi-tion Cgreaie + Cgr,,aih = 0 or CgreAe + CgrhAh = 0 for theuniform-plane-wave case. It is of practical importance tofind the condition maximizing Qimi and Qin2. Analysis ofEq. (113) with the help of Eq. (80g) reveals that if Ae and Ahsatisfy the following relation:

Ah/Ae = (Cgrh/Cgre) = P*, (114)

then the resulting input-coupling efficiencies Qim' and Qim2'take the following forms:

Qiml' = Qiel + Qihl (115)

Qim2 ' = Qie2 + Qih2- (116)

The maximum Qim2, max' of Qim2' is 81.45%, and the conditionleading to this maximum is given by Eq. (112b). The maxi-mum efficiency is the same as the one that was obtained forthe normal-incidence case.19 However, in order to get themaximum input-coupling efficiency by using a uniformplane wave for the oblique-incidence case, we generally haveto do one more task in addition to the adjustments of theincidence angles of the incident wave and the grating length.That task is forming the incident wave into an ellipticallypolarized wave whose rotation direction is the same as thatof the radiated wave from the output coupler as indicated inEq. (114).

Next, we consider an input coupler whose incident wave isthe time-reversed radiated wave of the output coupler. Thetime reversal is equivalent to taking the complex conjugateof the radiated wave; thus this type of input coupler is knownas a conjugate coupler. The input-coupling efficiency Q ofthe conjugate coupler for the principal guided wave Ag(Z2)propagating in the direction shown in the reciprocal opera-tion of Fig. 4 is defined as

IAg(0)I2 cos ,

IO, (Ai,1 + Aihl 2)dZ 2

We shall show that Qi, is identical to Q given by Eq. (96) orEq. (98), provided that Aie = bi,* and Aih = bih*. We manip-ulate the canonical equations for the output coupler given byEqs. (89)-(91) and the canonical equations for the reciprocaloperation given by Eqs. (86)-(88). Multiplying Eq. (91) byAg and Eq. (88) by ag, subtracting the latter result from theformer one, and using Eqs. (89) and (90), we find that

(117)

cos°g (Agag) = lbi l2-bih 2. (118)

Integrating Eq. (118) with respect to z2 from 0 to L and usingthe boundary condition Ag(L) = 0 yields

ALAg(Oag(O)cos g = | (Ibi, 12 + Ibih12 )dZ2- (119)

Substitution of Eq. (119) into Eq. (117) shows that Qi, andQot are identical. If the incident wave of the conjugatecoupler is of only one mode, say, Ai,, which is equal to b* ofthe output coupler, then the corresponding input-couplingefficiency Qics can be shown to be identical to Q given byEq. (97). Consequently, all the discussions relevant to theoutput-coupling efficiencies are valid for the input-couplingefficiencies of the conjugate coupler.

7. DESIGN PROCEDURES

In general, the reflection grating can be operated in fourdifferent schemes, namely, the corrugation on the conduct-ing surface with the principal guided wave of the TM mode(scheme 1) or of the TE mode (scheme 2) and the corruga-tion on the film-cover interface (Fig. 8) with the principalguided wave of the TM mode (scheme 3) or of the TE mode(scheme 4). The three canonical equations derived here forscheme 1 are formally valid for the other three schemes butwith different coefficients. A brief derivation of the coeffi-cients for the other schemes is outlined in Appendix A. Theseven properties given in Eqs. (80) are also valid for the otherschemes. Consequently, the proofs, the derivations, and theanalytical results contained in Sections 4-6 are general andcan be applied to all the four operating schemes.

In order for the reflection grating to function as an outputor an input coupler, a proper choice of the physical parame-ters of the reflection grating is necessary. Those parametersinclude the relative permittivities of the dielectric cover cand film Ef, the film thickness d, the grating wave number K,the propagation direction of the principal guided wave gthe modulation indices n1 and Y2, and the operating frequen-cy . All the parameters are normalized quantities, as be-fore. We initially assume that E = 1.0 and w = 1.0 except forthe situation when the frequency characteristics of the re-flection grating are under consideration. It is clear fromFig. 2 that the value of Og is between 0 and 900, and it is notnecessary for Og to be greater than 900. For simplicity, werestrict the parameters such that the radiated wave is con-fined in the region where x, y, and z are positive, whichmeans that the radiation angles 0 r and p, lie between 0 and90°. The requirement that there exist only one fast-wavemode leads to the following conditions (Fig. 3):

x

y z

EcCo, N x=d(z)

X=d

;tL A ff Hoe

Fig. 8. Geometry of the reflection grating with the periodic corru-gation on the film-cover interface.

Qi =


d32 + K2 - 2K1j cos Og < o 2 E,

0 2 + 4K2 - 4K13 cos Og > c 2

C'.

(120)

(121)

One convenient way to obtain a relation between 1j and K isto specify the radiation angle Or when Og = 00. Then weobtain from the definitions given in Eqs. (38a) and (100) that

K = j - cos r, Og = °

-Cggr

(122)

Substituting Eq. (122) into Eq. (121) with Og = 0° yields

1j > wf(1 + 2 cos 0,), (123)

which gives the lower limit of 1j. The upper limit of 13j canbe determined by fulfilling the following three restrictions:first, a chosen value of 13j together with K given in Eq. (122)must satisfy Eq. (120) for the largest possible value of Og orvice versa; second, since kj must be positive real by Eq. (23b)and Ef is a finite number, 1j must be smaller than JEf; third,in order for the waveguide to support only the fundamentalmode the film thickness d must be small such that cod(f -

e6)1/2 is less than ir for the TM-mode principal guided-waveoperations and less than 1.57r for the TE-mode cases, whichrestricts the lower limit of kj and hence the upper limit of Oj.After a value of 1j is chosen between the lower and the upperlimits and K is determined according to Eq. (122), the criti-cal angle Ogc, above which the radiation wave ceases to exist,is obtained from inequality (120) as

ogC = cot (i 2 K c).0gc = Cosi 2~

O I S [ o 1 30

Fig. 9. Leakage coefficient multiplied by -sec Og versus Og for thefour schemes. It changes substantially from one scheme to another,and for each scheme it maintains a nearly constant value up to Og S200. Oge is the cutoff angle above which the radiated wave does notexist. Scheme 1 has the largest value, and scheme 4 has the smallestvalue.

0.010-

C sec8g

(124)

The following design data meet all the above restrictions,and they are used in the subsequent sections to illustrate thecharacteristics of the output and the input couplers:

e = 1.0, Ef = 13 .0, c = 1.0,

0 r 600 whenOg = 0°, ql = 0.03, 12 = 0,

d(TM mode) = 0.4945, d(TE mode) = 0.7278,

Oj = 2.1000, K = 1.6000,

acj = 1.8466, kj = 2.9309,

00 < fig < g, (=27.330).

0.0000 30

8g[ 0 iFig. 10. Imaginary part for Cgg multiplied by -sec Og versus Og forthe four schemes. For 0 g 100, all the schemes have nearly thesame value of Cggi.

0 8[O] 30

Fig. 11. Radiation angles Or and oPr versus 0g. Note that Or becomessmaller while Opr becomes larger when Og increases and that the slopeof 'Pr with respect to Og is greater than unity for Og S 200.

coefficients on 0g. Perhaps the extinction coefficient Cgg isthe most important one, and its real and imaginary parts areplotted in Figs. 9 and 10. The dependences of the radiationangles Or and or are depicted in Fig. 11. The absolute valuesof the mode-conversion or mode-coupling coefficients areshown in Figs. 12 and 13. The absolute value and the argu-

100

[a](125)

All the other wave numbers can be calculated from thesedata. When we investigate the frequency characteristics ofthe reflection grating, we fix Ec, ef, K, d, and q as in Eq. (125)and vary w around 1.0, computing hj, 1j, and acj numericallyusing the corresponding dispersion relations and Eqs. (23).

08. CHARACTERISTICS OF THE OUTPUTCOUPLER

We shall illustrate the characteristics of the output couplerby using the design data chosen in Section 7 in the algebraicresults derived in Section 5, and we shall also give a briefcomparison among the four operating schemes. It should benoted that for the given normalized frequency co = 1.0 all thewave numbers are chosen to be the same for the fourschemes, and yet the coupling coefficients are different fromone scheme to another. Hence the evaluation of the cou-pling coefficients by using the design data should be carriedout first. Initially we investigate the dependences of the

(

(

.

1_.

W. S. Park and S R Seshadri

I


0.010-

ICgrI

aICgrhl2

0.000-0 g [ I] 30

Fig. 12. Mode-coupling coefficient ICgreI2 and mode-conversion co-efficient Cgrh12 versus 0

g for scheme 1 and scheme 3. We observethat, in general, mode-coupling coefficients become smaller whilemode-conversion coefficients become larger as 0g increases.

0.002-

- / Cgrel

0.000-0 e [3] 09II

Fig. 13. Same as Fig. 12 for schemes 2 and 4. Note that thecoefficients are much smaller than those in Fig. 12. In this figurelCgrh 12 is the mode-coupling coefficient and CgreI2 is the mode-con-version coefficient.

ment of the polarization factor P are shown in Figs. 14 and15.

From these figures, we observe several noteworthy fea-tures from which we can infer some aspects of the character-istics of the output coupler. It is noted from Fig. 9 that, foreach scheme, -Cggr sec Og maintains a fairly constant valueup to the cutoff angle gc, but each level differs substantiallyfrom one scheme to another. Since it is seen from Eq. (98)that the output-coupling efficiency is related directly to Cggrsee g, we anticipate that for a given output-coupling effi-ciency to be achieved, scheme 1 will require the shortestgrating length, whereas scheme 4 will require the longestgrating length. This fact is illustrated in Fig. 16, where Qtin Eq. (98) is plotted versus the corrugation length L for eachof the four schemes. As is seen from Eqs. (102), (103), and(106), the imaginary part Cggi of the extinction coefficient isa crucial part in determining the higher-order corrections ofthe radiation angles 0r and air. For convenience, we defineAOr', AOIr", and Ar' as

AOfI' = 0 / - ,/\r r Srs

AV ' = r - r

A(Pr = r - r-

(126a)

Now, we note from Fig. 10 that -Cggi sec Og for each scheme isapproximately the same in the range of g less than 100.This fact is reflected in Figs. 17-20, where the relative beha-viors of AOr', Ar", and Ar' for each scheme'are also shown tobe approximately the same for the range of Og. The depen-dences of 0 r and 'Pr shown in Fig. 11 are common for all theschemes since all the wave numbers are chosen to be com-mon. The data contained in Fig. 11 and their higher-ordercorrections, namely, Ar' and Arr' in Figs. 17-20, give the

5

IPI

0 I I . . I - J0 9g 1 ]30

Fig. 14. Absolute value of the polarization factor P versus Og, forthe four schemes. Scheme 2 and scheme 4 have almost the samevalue of IP throughout the entire possible range of g. The same istrue for schemes 1 and 3 for g 15°.

[x 1-0°° _ ,

Arg (P) 0 0

0.0-

-1.0- _.

0 g[] 30

Fig. 15. Argument of P versus Og for the four schemes. It can beproved that Arg(P) of scheme 1 is identical to that of scheme 3. Thesame is true for schemes 2 and 4.

1.0

0.<

(126b) ~~U L [A] 300(126b)[ ]Fig. 16. Total output-coupling efficiency Q versus the corruga-

(126c) tion length L. The unit of L is the spatial period A of the grating.

so

Vol..4, No. 6/June 1987/J. Opt. Soc. Am. A 1013

0.5- sum follow the same order as in Fig. 9 for the four differentschemes of operation. For the present range of g, there

- l does not exist an angle where the complete mode conversion

[ao ] ( ) 1/ occurs. When the corrugation length L is fixed at 50 A, theoutput-coupling efficiencies for the TM- and the TE-modecomponents of the radiated wave are illustrated in Figs. 21

and 22 for the four schemes. We notice that schemes 1 and 3

yield considerably higher total efficiencies throughout the

r possible range of Og. Some interesting properties associated

_AG68 ' with P are that, for a given mode of the principal guidedwave, IPI remains nearly invariant (Fig. 14) and Arg(P) does

0.0- not change (Fig. 15) for the two locations of the corrugation.

0 [ 30 The latter property can be deduced analytically from the9L

Fig. 17. Second-order corrections AO,.' and AOr/ and third-ordercorrection Ar" of the radiation angles versus Og for scheme 1. -

0.5 [a

[a] 0;5- . - . I [°]-ml-al

0° 30

0.0-0

6 r °1 30 DFig. 20. Same as Fig. 17 for scheme 4.

Fig. 18. Same as Fig. 17 for scheme 2. 1.0

0.5-

[o] o

Ar_. Hi k Ce 00A

0.0- l I . Fig. 21. Output-coupling efficiencies of the radiated wave of the

0 6 g [a] 30 TM- and the TE-mode types versus 0g for scheme 1 and scheme 3.

Fig. 19. Same as Fig. 17 for scheme 3.0.-

most accurate description of the propagation direction of theradiated wave from the present analysis.

As we mentioned above, the radiated wave, in general,consists of the TM- and the TE-mode components. In orderto understand the nature of the radiated wave and the rela-tion between the principal guided wave and those two com- 0Q

ponents in the radiated wave, we should know the mode-

conversion or -coupling coefficient Cgre and Cgrh, which arecomplex numbers. Instead, it is more convenient to consid-er ICgre12, ICgrhI2, P, and Arg(P), where the polarization fac- 0

tor P is defined in Eq. (99). ' We notice from Figs. 12 and 13 ( g l 30that ICgreI2 and ICgrh12 for'the TM mode are larger than those

for the TE mode and that the relative magnitudes of their Fig. 22. Same as Fig. 21 for scheme 2 and scheme 4.



0.90 1.00 1.10

Fig. 23. Spectral response of the radiation angle 0, for Og = 00. Theresponse for scheme 3 is identical to that for scheme 1. The same istrue for schemes 2 and 4. We note that 0

r decreases as w increases.

100

'l

0

0.90 1.00Wd

1.10

Fig. 24. Spectral responses of 0, and , for g = 100 for schemes 1and 3. We note that both 0 , and X, decrease as increases.

l00

[

0.90 1.00W

We shall now investigate the frequency characteristics ofthe reflection grating schemes by varying the operating fre-quency around &) = 1.0. The range of the operating frequen-cy is, for convenience, limited by the restrictions specified inSection 7. When the operating frequency is changed for thefixed geometry of the reflection grating schemes, all theresulting wave numbers and coefficients also change. First,we are interested in the spectral responses of the radiationangles and their higher-order corrections and of the polar-ization factor. Since these responses are also dependent on

0.5

[a)

0.0-0.90 1.00 I 1.10

Fig. 26. Spectral response of the third-order correction AO," forschemes 1 and 2 when Og = O'. The response for scheme 3 (4) isalmost identical to that for scheme 1 (2).

0.5

[o ]

I I I I 10.90 1.00 1.10

Fig. 27. Spectral responses of AX,' and AL," for scheme 1 when 0g =100. The responses for scheme 3 are almost the same as those forscheme 1.

1.10

Fig. 25. Same as Fig. 24 for scheme 2 and scheme 4.

definition of P. This tells us that the polarization of theradiated wave is almost unaffected when the location of thecorrugation is changed from the conducting surface to thefilm-cover interface or vice versa. It is obvious from Figs. 14and 15 that there does not exist a value of 0g for which theradiated wave is circularly polarized since P = 1 does notoccur when Arg(P) = ±7r/2. Hence, for the present range ofOg except when g = , all the radiated waves from the fourschemes are elliptically polarized. However, since Arg(P) isalmost zero for schemes 2 and 4 and -7r for schemes 1 and 3,the radiated waves are almost linearly polarized waves in therange of g less than 10°.

0.5 -

[] -

0.0-0. S)O 1.00

Fig. 28. Same as Fig. 27 for scheme 2. The responses for scheme 4are almost the same as those for scheme 2.

2 e = 8g =00}~ ~ ~ ~~( - (3

a = oa

I I I

0 9=10°A I g Or

A er

-1

IsA r

l l |

I

-

I1.10


5

IPI

0.90 1.00 1.10

Fig. 29. Spectral response of IPI for scheme 1 and scheme 2 when Og= 100. The response for scheme 3 (4) is almost identical to that forscheme 1 (2).

1.0_

[x180 a]

A rg (P)

0.0- .4

-'.0 --0.90

I I1.00 l.10W

Fig. 30. Spectral response of Arg(P) for the four schemes when Og =100

I .C

O.c

0.90 I.00WA

1.10

Fig. 31. Spectral response of the total output coupling efficiency

Qot for the four schemes when 0g = 0° and L = 50 A. Note that Qot

decreases as co increases.

Og, we consider two typical values of Og, namely, 0° and 100.

We shall consider only scheme 1 and scheme 2 because

scheme 3 (scheme 4) satisfies the same dispersion relation as

scheme 1 (scheme 2), resulting in similar or identical spectralresponses. Figures 23-25 show the dependences of 0r and 'r

on co, and we note that these two angles decrease as co in-

creases. The radiation-angle corrections Air" and A'r' ver-sus co are plotted in Figs. 26-28. We notice that these cor-rections are indeed smaller than 0r or 'pr by 2 orders for theentire allowed frequency range. The polarization factor P

0.

has no meaning when 0g is equal to zero. The spectralresponses of IPI and Arg(P) for Og = 10° are shown in Figs. 29and 30. We note that P1 undergoes a substantial changewhile Arg(P) remains almost constant as co varies. Next, we

shall consider the dependence of the output-coupling effi-ciencies on the operating frequency for a given corrugationlength L = 50 A. One common phenomenon is that the total

output-coupling efficiency Qot always decreases as co in-creases. When Og = 00, Q0t versus c for the four schemes is

shown in Fig. 31, where the order of magnitude is the same asthat in Fig. 9. We know from Fig. 29 that for fIg = 100 therange of IPJ varies significantly across either side of unity asc is changed from 0.9 to 1.1. This result is reflected in theoutput-coupling efficiencies Qoe and Qoh, of which the domi-nant one changes over the frequency range, as shown in Figs.

32 and 33.

9. CHARACTERISTICS OF THE INPUTCOUPLER

For the given geometry of the reflection grating, the input-coupling efficiencies derived in Eqs. (110), (111), (115), and(116) depend on three variables, namely, the corrugationlength L, the operating frequency w, and the direction of theincident wave. The dependences of Qim2' on L for Og = 100,with the incident angles fixed at their optimum values 0i' andXi', are plotted in Figs. 34 and 35. We note that for each

scheme Qir2,max' (=81.45%) is achievable, yet the required

1.0-

Q h

0.0-10.90 1.00 1.10

Fig. 32. Spectral responses of Qee and Qoh for schemes 1 and 3 whenOg = 100 and L = 50 A. Note that as w increases, Qoe and Qoh undergoa substantial change, while their sum decreases slightly.

90 1.00

Fig. 33. Same as Fig. 32 for schemes 2 and 4.

1.10

-~~~ I

j~~

- O0

4



L [A]Fig. 34. The input coupling efficiency Qi.2' at the optimum inci-dent angles versus L for schemes 1 and 3 when Og = 100. Note thatthere is an optimum corrugation length Lop for each scheme.

l.0-

Qim2

0.0- I

of Qie2 and Qih2 on L take the same graphical representationsas Qir2' given in Figs. 34 and 35, but the scales in the ordin-ates should be properly marked by using the relation Qih2/Qie2 = JP12, where P is given in Fig. 14.

When the spectral response of the input-coupling efficien-cy is under consideration, the incident angles and the corru-gation length are fixed at their optimum values for the fre-quency c = 1.0. Figure 36 illustrates Qim2 versus for g =0° for the four schemes. We notice that the frequencyselectivity, defined as the ratio of the center frequency to thehalf-power bandwidth, differs substantially from onescheme to another, and it is proportional to the optimumlength Lops Table 1 lists the selectivity and Lop for eachscheme.

Finally, we investigate the dependence of the input-cou-pling efficiency Qimt' on the deviations of the incident anglesfrom their optimum values. For each scheme the corruga-tion length is fixed to Lop and the operating frequency is 1.0.In order to describe the incident angles for the incident wavegiven in Eq. (109), it is necessary to define il and il asfollows:

=il arctan[k_,/(A0 sin' 0 g + 1 3 2)i]

il = arctan(3j sin Og/lo),

(127)

(128)

where

1 = j cos Og-K + (1 + q)Cggi sec 0g (129)

It is seen from these definitions that i (=r), 'Pi (r), i'(=Or'), and i' (r') are special cases of il and il as

Fig. 35. Same as Fig. 34 for schemes 2 and 4.

0.0-f--2.5 0.0 2.5W-1 [x I3]

Fig. 36. Spectral response of the input coupling efficiency Qim2'with the corrugation length and the incident angles adjusted to theiroptimum values for = 1.0 for each scheme. A graphic measure-ment of the half-power bandwidth for scheme 1 is illustrated.

optimum corrugation length Lp is different from onescheme to another according to the design formula given byrelation (113). It is seen from Fig. 9 and relation (113) thatfor each scheme Lop remains nearly constant for the range ofOg less than 20°. The optimum angles Oi' and 'i' can beobtained graphically from Figs. 11 and 17-20 and by usingthe definitions given in Eqs. (126a) and (126b). Lop is short-est for scheme 1 and longest for scheme 4. The dependences

0i = Oil(q = -1),

Xi = Oil(q = -1),

(130a)

(130b)

Table 1. Correlation between the FrequencySelectivity and the Grating Length for the Four

Operating Schemesa

Scheme1 2 3 4

Selectivity 355 2470 603 9500Lop (A) 80 677 137 2627

a Note that the selectivity becomes larger as the grating becomes longer.

l.0-

Q I - eg~~~ =00Qiml gD

0.0--15 0 15

qFig. 37. The input coupling efficiency Qi I' for 0g = 00 versus theintermediate variable q with the operating frequency fixed at 1.0and the corrugation length set to Lop for each scheme.


0q

Fig. 38. The incident angle Oil versus the intermediate variable qfor all the schemes when Og = 00. Note that the slope of il withrespect to q is approximately the same for all the schemes.

55

I 0J

[a]

52-t-I5 0

qFig. 39. Same as Fig. 38 for Og = 100.

0q

15

Fig. 40. Same as Fig. 39 for Pil versus q.

0i' = 0il(q = 0), (131a)

hi' = 'il(q = 0). (131b)

The fact that the canonical equation (72) can deal withincident waves whose phase is only of the type given in Eq.(109) implies that it can provide solutions for the input-coupling efficiency only for the incident wave whose incidentangles are represented by Eqs. (127)-(129). For a given fIg,when q varies, 0il and 'il form a surface in the three-dimen-sional space. For example, if Og = 00, the xz plane is thesurface. Thus we allow the deviations of the incident anglesonly on the surface. In relating Qimi' to the two angles Oil

and il, we use q as an intermediate variable. Figure 37shows Qiml' versus q for g = 0°. Since Lop is different foreach scheme, it is expected from the definition of Qirni' thatscheme 4, whose Lop is the largest, will be the most sensitiveto q and that scheme 1, whose Lop is the smallest, will be theleast sensitive to q. The behavior of Oil depends on Cggi.Since Cggi is nearly the same for the four schemes (Fig. 10),Oil versus q does not change significantly from one scheme toanother, as shown in Fig. 38. However, since it is seen fromFig. 10 that -Cggi sec 0 g for schemes 1 and 3 is slightly largerthan those for other schemes, the slope for schemes 1 and 3in Fig. 38 is slightly steeper than those for schemes 2 and 4.When Og = 100, the input-coupling efficiency Qirni' versus q isalmost identical to that in Fig. 37 for the four schemes; henceit is not drawn. However, we need two figures, Figs. 39 and40, to describe the loci of the incident angles Oil and 'il whenq varies. Again, in both figures, the slopes for schemes 1 and3 are a little steeper than those for schemes 2 and 4 because-Cggi sec Og for schemes 1 and 3 is slightly larger than thosefor schemes 2 and 4, as shown in Fig. 10.

10. CONCLUSIONS

A reflection grating can be designed by incorporating a prop-er periodic corrugation in a grounded dielectric film wave-guide, and basically it is operated in four different schemes.We have investigated the interactions between the guidedand the radiated waves in the reflection grating for theoblique-incidence case, where the wave vector of the guidedwave is inclined to the grating vector. Three generally validcanonical equations are rigorously derived, and all the cou-pling coefficients for the four schemes are explicitly ex-pressed in terms of the physical parameters of the reflectiongrating. Seven properties satisfied by the coupling coeffi-cients are deduced, and these lead to the power-conservationprinciple and reciprocity. One feature pertaining to theoblique-incidence case is that the radiated wave and theincident wave include both the TM- and the TE-mode com-ponents, regardless of the mode type of the principal guidedwave.

Based on the solutions of the canonical equations and withthe help of the first-order field solutions, we have derivedthe following expressions for the output coupler: the out-put-coupling efficiencies for the individual radiated mode aswell as for the total radiated wave that give the portion of theenergy in the principal guided wave that is transferred to theradiated wave; the polarization factor that describes thetype of polarization of the total radiated wave; and the rera-diation angles and their corrected versions, taking into ac-count second- and the third-order analysis.

In the input-coupler analysis, the canonical equation cangive a solution for the input-coupling efficiency only for theincident wave whose amplitude is a function of the slowvariable employed in the multiple-scale expansion. Theincident wave formed by a uniform plane wave of only onepolarization or a mixture of the two polarizations is consid-ered. Analytical expressions for the optimum incident an-gles are obtained. For the optimum incident angles, themaximum input-coupling efficiency for the incident wave ofone polarization is achieved for a particular value of thegrating length. Analytical expressions for the maximuminput-coupling efficiency and the optimum grating length

W. S. Park and S. R Seshadri


are also deduced. If the incident wave is a mixture of thetwo polarizations, one additional condition for achievementof maximum input-coupling efficiency is that the polariza-tion of the two incident modes be the complex conjugate ofthat for the total radiated wave of the output coupler. Wehave also shown that the input-coupling efficiency of theconjugate coupler takes the same form as the output-cou-pling efficiency of the output coupler.

For a given geometry of the reflection grating, all thepreviously mentioned expressions are dependent on the in-clination angle g and the operating frequency . For eachof the four schemes one example of the reflection grating istreated to illustrate the dependences and to make somecomparisons among the schemes. From these numericalresults we can draw the following general conclusions:Scheme 1 has the definite advantage of requiring the shor-test corrugation length to achieve a given output-couplingefficiency and to obtain the maximum input-coupling effi-ciency, and scheme 4 requires the longest corrugation lengthfor the same purpose. However, the input-coupling effi-ciency for scheme 4 has the greatest frequency selectivityand is the most sensitive to the variation of the incidentangles around the optimum values. For a given corrugationlength and for a given scheme, as the inclination anglechanges, the output-coupling efficiencies for the individualmodes change significantly, but the total output-couplingefficiency remains almost constant. The sagittal planeformed by the radiation vector and the normal vector to thereflection grating is more inclined to the grating vector thanthe plane formed by the wave vector of the principal guidedwave and the normal vector. This implies that for a givenprincipal guided wave a change in the direction of the grat-ing vector results in a correspondingly bigger change in theradiation direction. The total radiated wave is always anelliptically, but close to linearly, polarized wave. Whetherthe corrugation is located on the conducting surface or onthe film-cover interface does not affect the polarization ofthe total radiated wave, provided that the principal guidedwave is of the same mode (TE or TM). As the operatingfrequency increases, the radiation angles and the total out-put-coupling efficiency decrease. The input-coupling effi-ciency is extremely sensitive to the variations of the operat-ing frequency and the incident angles, especially for thescheme requiring longer optimum corrugation length.

APPENDIX A

Coupling Coefficients for Scheme 2The coupling coefficients of the canonical equations forscheme 2 can be derived by using the same procedures asthose for scheme 1. The boundary conditions remain thesame for all the orders a" (n = 0, 1, 2). The only nonvanish-ing x component of the zero-order field is HO, and its solu-tions are expressed as

H = Nghag(z 2 )exp(i01)exp[-aj(x - d)]

X exp[i(j sin Ogy + j cos Ogzo)], d < x < , (Al)

H. = Ng ,ag(z 2)exp(i01 ) i(hid)9 exp~i ~ smn( d)

X exp[(13 sin g + 3j cos Ogzo)], 0 < < d. (A2)

The zero-order dispersion relation is obtained as

acj = -kj cot(kjd).

P 3 =

P 4 =

(A3)

In order for Pgz to take the same form as in Eq. (33), thenormalization constant Ngh should be chosen as

(A4)Nh= [ 40ja jk1 1/2

The first-order fields include both E 1 and H.1 as before, andtheir governing equations are given by Eqs. (34)-(37) withthe proper choice of sign in front of bie and bih for conve-nience. The applications of the first-order boundary condi-tions lead to the canonical equations (45) and (46) with thefollowing coefficients:

Crge = dl1le,,fKkj sin OgNgh213 sin(kjd)cos(k_ 1 fd) [efk.l - ik-lf tan(k-lfd)]Nre

(A5)

Crgh = d711 kj( - K cos Og)k-IfNgh23j sin(kjd)sin(k_1fd)[ik_1 - k cot(k-lfd)]Nrh

(A6)

The expressions for Crre and Crrh are the same as those givenin Eqs. (48) and (50), and they are valid for all the schemes.

To determine the solvability condition for the second-order problem, we assume particular solutions of E 2 and Hy2in the form given by Eqs. (59) and (60). One fundamentaldifference in the procedures of obtaining the solutions for V/'xrand O.,r (r = f, c) and of applying the second-order boundaryconditions to these solutions stems from the fact that thezero-order fields in scheme 1 are different from those inscheme 2. The amplitude transport equation takes thesame form as Eq. (72) but with different coupling coeffi-cients, as follows:

C = 1 (E+ dac1) (P1 + P 2 + P 3 + P 4), (A7)

p = - 2 ef 2 sin2 0g[ieck-1f cot(klfd) + fk 1 ]

k-lf[iEklf - kIc cot (k-1 1 d)] (2 + K2 - 21jK cos Og)

(A8)

( - K cos g)2k_1f[ik_1f tan(k-lfd) -k-c](flj2 + K2 - 2 jK cos Og) [ik-lf + kc tan(k 1 fd)]

(A9)

c 2EfTi2 sin 2 Og[Ecfalc tanh(alfd) + Efalf]

al(j/ + K2 + 2fljK cos Og) [efalc + ecalf tanh(afd)]

(A10)

-(# + K cos g)2a lf[alf tanh(alfd) + a](1j2 + K2 + 2jK cos Og) [alf + a tanh(afd)]

Cgre = d 1cve0 efK sin Og cos(kjd)kj 2k-1,Nre(0j2 + K2 - 213jK cos 0g)(l + daC)cos(k_1fd)Ngh

X [ieck-lf tan(klfd) - Efk(A),


(A12)


dilk 2 cos(kJd)k-f(j - K cos Og)klcNrh

grh (1 + da)sin(kifd)(132 + K 2 - 2jK cos Og)Ngh

X klf cot(k-lfd) - ik-c 1]. (A13)

Coupling Coefficients for Scheme 3 and Scheme 4The governing equations for schemes 3 and 4 are the same asthose given in Eqs. (8)-(17). However, since the periodiccorrugation, represented by

d(z) = d[l + 577i cos(Kzo + 1)

+ 62n2 cos(2Kzo + 2) + . . , (A14)

is located on the film-cover interface, as shown in Fig. 8, thecorresponding boundary conditions, which are valid forschemes 3 and 4, are completely different from those forschemes 1 and 2. For each order a/n (n = 0, 1, 2), En and Eznvanish on the x = 0 plane. By using Taylor series expansion,the equivalent boundary conditions at x = d for each order anare obtained as follows:

0(0):

Uy0, UzO: continuous,

Uy, + d%1 cos(Kzo + °1) a UY0,

Uzj + dql cos(Kzo + 1)a U;o-O

- dq1K sin(Kzo + fI1) Uxo: continuous.

(A15)

(A16a)

(A16b)

Uy2 + d771 cos(Kzo + 1) a- UY1 + dn2 cos(2Kzo + 2) a UYOax -a - Uy

+ d212 cos2(Kzo + f1) U,

2 a x2 ~0(A17a)

U22 + dq, cos(Kzo + 01) a UZi

+ d72 cos(2Kzo + 02) a U210 - dK sin(Kzo + f 1)Uxlax

+ 2 d2 72 cos2(KzO + 1) 2 Uzo2 Ox2 zO

- d2 ij12K cos(Kzo + fI1)sin(Kzo + fI1)a Xax

- 2d- 2K sin(2Kz 0 + 2) Uxo: continuous, (A17b)

U = E or H.

The zero-order solutions for scheme 3 and scheme 4 areidentical to those for scheme 1 and scheme 2, respectively.In addition, the subsequent procedures leading to the canon-ical equations for scheme 3 (scheme 4) are similar to thosefor scheme 1 (scheme 2). The resulting coupling coefficientsfor scheme 3 are as follows:

Crgednl(Ef - EC)Nge

2#j[iefk.lC + Eck-lf tan(k.lfd)]Nre

X Kj(j cos Og - K) - j - K cos Og)

X [aCjk-lf tan(klfd) + 1j32]j,

-dj(Ef - e)cvaK sin OgNge

rgh 21j[iklf cot(k-lfd) + k 1c]Nrh

Cg = 4D (Ef c kaj(Pl + P2 + P3 + P4),

(A18)

(A19)

(A20)

P1 = {-ef(ef - ec)a'I 2k-lk.lf(flj -K cos Og)2 tan(klfd)

+ eCEfaCk-lk.lf(0 -K cos Og)2

x [-k-lf + ik-1, tan(k-lfd)]- iE,(Ef - )1j 2(13j2 + K2 - 2Kj cos g)2

+ ecefaC(p 2 - K2 cos2 g)(3j2 + K2 - 21jK cos Og)

X [k-, - ik-1 ftan(k_,fd)]}X (0/ + K2 - 2Kj cos 0g)[iEcklf tan(k-lfd)-EfklC]} ,' (A:

W2EfajK2 sin2 g

2 =( 2 + K2 - 2Kj cos Og)

x [E ~ + i(fEf - c)aj ] (Aik1 1fcot(k 1 1fd) + k-Ic

P3 = 1Ef(Ef - E,) aj2a1aa(j + K cos Og)2 tanh(afd)

+ ECEfatCjajcaj(j + K cos Og)2[alf + aic tanh(alfd)]

- Ej~f - E )1j2(1j2 + K2 + 21jK cos og)2

+ cEfa c(0/ 2- K2 cos2 Og)(3j2 + K2 + 21jK cos Og)

X [alc + alftanh(alfd)]I f(1j2 + K2

+ 2Kj cos Og) [-EfalC - calf tanh(atfd)]Ij1, (A,

co2efacjK2 sin2 0g

(32 + K2 + 2Kj cos g)

X > +(cf -E) 1

I,alf coth(a 1 f1 d) + a,,

Cgrs = idnlEcEf(Ef - Ec)ajk 2k-.i[(OI - K cos Og)acjklf

X tan(k-lfd) + 1j(1j2 + K2 - 2K1j cos fg)]

X N -[ Efk- iekk-lf tan(k. 1fd)]

X (13 + K2 - 2K1j cos Og)DNge 1-',

Cgrh =

(A24)

(A25)

-dnliwvE$Ef - E)K sin fgkj2aj 2kleNrh

[ik-lf cot(k.lfd) + k-1 ,] (1j2 + K2 - 2K13 cos fg)DNge

(A26)

The coupling coefficients for scheme 4 are obtained as fol-lows:

(A27)Crge = dfl-c(ef - E,)K sin Ogk-lf tan(klfd)Ngh

20j[-fk-, + i k.lf tan(k.lfd)]Nre

21)

.22)

23)



idn (f3j - K cos g) (kj2 + acj2)NghCrgh -2fljIh. 1 , + i- 1f cos(khlfd)]Nh(A8

=C+ (P + P2 + P3 + P4), (A29Cg=413j(1 + dacj)

P, = K' sin' 0gaj-ekj + ick-I.f tan(k.lf1 d)J+ (Ef - Ec)kjkf tan(klfd)jX : j2 + K2 - 2Kj cos g)

X [-cfkjc + ieCk.lf tan(klfd)]P-1, (A30,

P2 = Wj3 - K COS fOg)2 lcj[k..lc + ik...1 cot(k.l1fd)]

- i 2(ef - Ec)II(0/2 + K2 - 2Kj cos g)

X [k-lc + ik..f cot(kjf]d1V 1, (A31)

P3 =K K2 sin 2 f0gkaci[fE~ec + ecaif tanh (a1fd)]

+ ( - e)aeajf tanh(afd)1

X (j2 + K2 + 2Kj cos Og)

X [al, + Ealt tanh(a1 fd)] , (A32)

P4 = (j + K cos Og)2 lacj[a la + alf coth(alfc)] - W2(ef -Ec)

X (132 + K2 + 2Kj cos g)[ac + alf coth(a1fd)]) ,

(A33)

Cgre = dl eck, 2aj 1K sin gk-.ik-lf tan(k-ld)Nre

X c( + da j)(132 + K2 - jcosf g)

X [-cfk.C + ick-lf tan(kjlfd)INghP1 , (A34)

C91h =ridia jkk-lc(#j - K cos Og)Nrhgrh (1 + dacj)[k-lc + ik-If cot(k_ld)](1j2 + K2 - 2Kj cc

tion of the coupling coefficient of a holographic thin film cou-pler," Opt. Commun. 9, 240-245 (1973).

5. A. Jacques and D. B. Ostrowsky, "The grating coupler: com-parison of theoretical and experimental results," Opt. Commun.13, 74-77 (1975).

6. V. A. Kiselev, "Diffraction coupling of radiation into a thin-filmwaveguide," Sov. J. Quantum Electron. 4, 872-875 (1975).

7. D. G. Dalgoutte and C. D. W. Wilkinson, "Thin grating couplersfor integrated optics: an experimental and theoretical study,"Appl. Opt. 14, 2983-2998 (1975).

8. C. C. Ghizoni, B. U. Chen, and C. L. Tang, "Theory of experi-ments on grating couplers for thin-film waveguides," IEEE J.Quantum Electron. QE-12, 69-73 (1976).

9. A. Yariv and M. Nakamura, "Periodic structures for integratedoptics," IEEE J. Quantum Electron. QE-13, 233-253 (1977).

10. T. Tamir and S. T. Peng, "Analysis and design of grating cou-plers," Appl. Phys. 14, 235-254 (1977).

11. P. K. Tien, "Method of forming novel curved-line gratings andtheir use as reflectors and resonators in integrated optics," Opt.Lett. 1, 64-66 (1977).

12. K. Wagatsuma, H. Sakaki, and S. Saito, "Mode conversion andoptical filtering of obliquely incident waves in corrugated wave-guide filters," IEEE J. Quantum Electron. QE-15, 632-637(1979).

13. A. Hardy and W. Streifer, "Analysis of waveguided Gaussianbeams coupled by misaligned or curved gratings," J. Opt. Soc.Am. 69,1235-1242 (1979).

14. P. K. Tien and R. J. Capik, "A thin film spectrograph for guidedwaves," in Digest of Topical Meeting on Integrated and Guid-ed- Wave Optics (Optical Society of America, Washington, D.C.,1980), pp. TuB3-1-TuB3-4.

15. S. R. Seshadri, "TE-TE mode coupling at oblique incidence in aperiodic dielectric waveguide," Appl. Phys. 25, 211-220 (1981).

16. S. R. Seshadri and M. C. Tsai, "Mode conversion of obliquelyincident guided magnetic waves by a grating on a yttrium irongarnet film for the normal magnetization," J. Appl. Phys. 56,501-510 (1984).

(A35)

ACKNOWLEDGMENT

The research of Wee Sang Park was supported in part by theEngineering Experiment Station, College of Engineering,University of Wisconsin-Madison.

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20. A. Neyfeh, Perturbation Methods (Wiley, New York, 1973),Chap. 6, pp. 228-307.

21. W. S. Park and S. R. Seshadri, "Reradiation from a gratingcoupler for a grounded dielectric slab waveguide," Proc. Inst.Electr. Eng. Part H 133, 10-17 (1986).

22. W. S. Park and S. R. Seshadri, "Higher order characteristics ofwaves on a sinusoidally modulated reactance surface," Electro-magnetics 4, 89-109 (1984).


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Reflection Grating Couplers at Oblique Incidence